High-dimensional limit theorems for random vectors in ${\ell }_p^n$-balls

Abstract

In this paper, we prove a multivariate central limit theorem for ${\ell }_q$-norms of high-dimensional random vectors that are chosen uniformly at random in an ${\ell }_p^n$-ball. As a consequence, we provide several applications on the intersections of ${\ell }_p^n$-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an ${\ell }_p^n$-ball onto a line spanned by a random direction $\theta \in {\mathbb{S}}^{n-1}$. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime $1\le p<q$ this displays in speed and rate function deviations of the $q$-norm on an ${\ell }_p^n$-ball obtained by Schechtman and Zinn, but we obtain explicit constants.

References in corpus (1)

• A probabilistic approach to the geometry of the \ell_p^n-ball